Statistics Take-Home Activity: Probability Distributions, Mean, Standard Deviation, and Sampling

Question 1: Probability Distributions

1.The time, in minutes, that a student needs to complete this Take-Home activity is uniformly distributed between 20 and 40 minutes, inclusive.

a.What is the probability that a student needs fewer than half an hour?

b.On the average, how long must a student need to finish this activity? 

c.Find the mean, µ; and the standard deviation, σ.

d.Ninety percent of the time, the time a student will need to finish this activity falls below what value? Note: P (X < k) = 0.90 where k is some value in the distribution range. 

e.Find:  P (20 ≤ X ≤ 35 or 30 < X < 45) = ?

2.)Let X be a normally distributed variable with mean; 30, and standard deviation; 4. Find the values of these probabilities:

3.)Suppose a class has 20 students (to begin with), that each student drops independently of any other student with a probability of 0.3. Let X represents the number of students that finish this course.

a.What kind of distribution is this describing? Explain?

b.Find the probability that X is between 12 and 14 inclusive. (Hint: use binomial table)!

c.an approximation appropriate for the number of students that finish the course?

d.If so, what is this distribution and what are the value(s) of its parameter(s)?

e.Find the probability that is between 12 and 14 inclusive by using the approximation (if an approximation appropriate)

f.Now, if a student is in the top 5% of whom finish the course, what is the minimum number of students who finish the course?

4.)Suppose a simple random sample of size 50 is selected from a population with σ = 10. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). 

a.The population size is infinite. 

b.The population size is N = 500. 

5.)To estimate the mean age for a population of 4000 employees, a simple random sample of 40 employees is selected where the population standard deviation is σ = 8.2 years 

a.Would you use the finite or infinite population correction factor in calculating the standard error of the mean? Explain. 

b.What is the probability that the sample mean age of the employees will be within 2 years of the population mean age? 

c.A market research firm conducts telephone surveys with a 49% historical response rate. What is the probability that in a new sample of 600 telephone numbers, at least 175 individuals will cooperate and respond to the questions?

7) What is the relationship between sample size and standard error?

8.) A population proportion of the active COVID-19 cases in mild condition is 0.95 (from last update on March 30, 2020 at 21:26 GMT “ www.worldometers.info “). A simple random sample of size 7000 cases will be taken and the sample proportion  will be used to estimate the population proportion.

a.What is the probability that the sample proportion will be within  of the population proportion?

b.What is the probability that the sample proportion will be within  of the population proportion?